Fourier coefficients of cusp forms and automorphic $L$-functions

O. M. Fomenko

Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 14, Tome 237 (1997), pp. 194-226

Résumé

In the beginning of the paper, we review the summation formulas for the Fourier coefficients of holomorphic cusp forms for $\Gamma$, $\Gamma=\operatorname{SL}(2,\mathbb Z)$, associated with $L$-functions of three and four Hecke eigenforms. Continuing the known works on the $L$-functions $L_{f,\varphi,\psi}(s)$ of three Hecke eigenforms, we prove their new properties in the special case of $L_{f,f,\varphi}(s)$. These results are applied to proving an analogue of the Siegel theorem for the $L$-function $L_f(s)$ of the Hecke eigenform $f(z)$ for $\Gamma$ (with respect of weight) and to deriving a new summation formula. Let f(z) be a Hecke eigenform for $\Gamma$ of even weight $2k$ with Fourier expansion $f(z)=\sum^\infty_{n=1}a(n)e^{2\pi inz}$. We study a weight-uniform analogue of the Hardy problem on the dehavior of the sum $\sum_{p\le x}a(p)\log p$ and prove new estimates from for the sum $\sum_{n\le x}a(F(n))^2$, where $F(x)$ is a polynomial with integral coefficients of special form (in practicular, $F(x)$ is an Abelian polynomial). Finally, we obtain the lower estimate $$ L_4(1)+|L'_4(1)|\gg\frac1{(\log k)^c}, $$ where $L_4(s)$ is the fourth symmetric power of the $L$-function $L_f(s)$ and $c$ is a constant.